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Theorem psubclssatN 38750
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a 𝐴 = (Atoms‘𝐾)
psubclssat.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclssatN ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3 𝐴 = (Atoms‘𝐾)
2 eqid 2733 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 psubclssat.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3psubcliN 38747 . 2 ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
54simpld 496 1 ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3947  cfv 6540  Atomscatm 38071  𝑃cpolN 38711  PSubClcpscN 38743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-psubclN 38744
This theorem is referenced by:  pmapidclN  38751  psubclinN  38757  paddatclN  38758  pclfinclN  38759  poml6N  38764  osumcllem3N  38767  osumcllem9N  38773  osumcllem11N  38775  osumclN  38776
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